I might be splitting semantic hairs, but my view is that it's not exactly correct to say Aikido "has no spirituality". I would say Aikido has no religion. Certainly Aikido is many things to many people and in that sense it won't always have spirituality attached to its practice, but in my own case I certainly derive spiritual practice from my Aikido practice...and that has nothing to do with my practice within the Shinto religion. I think OSensei intended Aikido to be spiritually enriching.
Spirituality resides in the person. Everyone's spirituality is unique and it is with them whatever they are doing.
It can be separate from religion but most people use a doctrine or code of beliefs to understand and express their spirituality.
Any physical activity can be used as as a tool to spiritually enrich, depending on the state of mind of the person when doing the activity.